TWISTORS AND GAUGE FIELDS

We describe briefly the basic ideas and results of the twistor theory. The main points: twistor representation of Minkowsky space, Penrose correspondence and its geometrical properties, twistor interpretation of linear massless fields, Yang-Mills fields (including instantons and monopoles) and Einstein-Hilbert equations.


I. INTRODUCTION.
The theory of twistors, first initiated by R. Penrose at the end of the sixties, now enjoys a period of rapid development.The increased interest in this theory is explained, mainly, by its applications to solving the fundamental nonlinear equations of theoretical physics, namely Yang-Mills and Einstein-Hilbert equations (YMand EH- equations, for short).The twistor method together with the closely related inverse scattering method have now become the main methods available to construct classical solutions of these equations.In this review we describe briefly the basic ideas and results of the twistor theory.
For a more detailed exposition and further references see [I-3].
Twistor model of Minkowsky space.The Minkowsky space-time M which is the basic space of relativistic field theory appears in the twistor theory in a rather unusual form.The twistor model of M is constructed in two steps, the first of assigns to every point x e M an Hermitian 2 2 -matrix X.The space of all such matrices is called the spinormodelofMinkowskispace and is denoted by the same letter M.
The vector space S C 2 where these matrices operate, is called the spinor space.A 0 The coordinates of spinors, i.e. vectors z S are denoted by z (z z ); A coordinates of complex conjugate spinors, i.e. vectors w e S, are denoted by w (w 0' w ).Dual spinors, i.e. vectors u S', v S', are denoted: u A (u 0, Ul), VA, (v0,, v I,)-AA' In spinor notation the mapping (i.i)has the form: x (xa) X (x where A 0,I Using the standard basis in the space of Hermitian 2 2 -matrices: AA' where o are entries of the matrix o a 0,I,2,3 in spinor notation.
a a Physicists usually omit the "Z" sign in formulas and sum over repeated indices.
The eton of the Lorentz group L (the group of orthogonal transformations of M in Lorentz metric) corresponds to the inner action of the group SL(2,C) (2x2 -matrices with determinant i) on Hermitian matrices: X AxA* A SL(2,C).The group SL(2,C) is thus seen to be the two-fold covering of the identity component L of O the Lorentz group (note that elements A and -A of SL(2,C) induce the same trans- formation of M).In other words, the spinor space is a space of the two-dimensional (fundamental) representation of the group SL (2,C)   the two-fold covering of the Lorentz group.The mapping (I.i) can be extended to the complex Minkowski space CM, which is a space of points z (z a) C 4 by the following formula: a The image of this mapping coincides with a space of all complex 22 -matrices which is called the spinor model of CM (and is denoted by the same letter), the norm Izl 2 (gO) 2 (zl) 2 (Z2) 2 (Z3) 2 transforms into det Z.
Let's turn now to a construction of the twistor model of Minkowski space.
Assign to a matrix Z CM the complex 4x2 -matrix (-iZ) where 12 is the iden-

AA'
=> In other words, we assign to a matrix (z the plane defined by a pair A AA' f linear equations: -lZ hA' Since these equations are homogeneous in (oA,nA,) they also define a proective line in the 3-dimensional complex projective space CP 3. The space T C 4 with coordinates (A,A,) and the corresponding space PT CP 3 are called the spaces of twistors an._ective twistors respective- ly.
The superposition mapping A ' (z a) {(A,A, ): m -iz A' (1.4) asigns to a point z (z a) of the complex Minkowski space CM a two-dimensional plane in the twistor space T or a projective line in the space PT.The mapping (1.4) extends also to the conformal compactification CM of the complex Minkowski space CM, which is obtained by "adding" to CM a complex light cone "at infinity" (cf. [2]).
We have defined the mapping (1.4) which assigns to every point of CM a projec- tive line in PT The associated correspondence between points of CM and PT is called the Penrose correspondence.We now consider geometric properties of this cor- respondence.These properties are formulated in diagrams where geometrical objects of CM occupy the left side and corresponding objects of PT the right side.We have: {point of CM} {projective line of PT The converse assertion: ig" i =>{null complex 2-planei point of PT bundle of projective in CM a-plane lines passing through a fixed point of PT A null plane is by definition, a plane such that the distance in the complex Lorentz metric) between any two of its points is zero.A null plane corresponding to a point of PT by the Penrose correspondence is called an a-plane.The dual asser- tion: [.=> {null complexin CM

2-plane B-plane
The "intersection" of the last two diagrams gives: projective plane of PT E point of PT* pencil of projective lines lying in a fixed projective plane null line in CM complex light ray It follows from the last assertion: complex light cone in CME bundle of complex light rays passing through a fixed point of CM (0,2)-flag in PT E (point of PT projective plane in PT including this point) E pair of incident points of PT and PT* projective line of PT E (0,1,2)| -flags in PT with a fixed projective line These are the main facts of the twistor Beometry for the complex Minkowski space.
Let's now consider the real compactified Minkowski space M. Denote by N the quadric in T given by the equation: (Z) IZ012 + IZII 2 IZ212 IZ312 0, and let PN be the associated projective quadric in PT.A restriction of the Penrose correspondence to M has the following properties: -ig"5] {point of M null line of M light ray of M light cone of M bundle of light rays passing through a fixed point of M x + iye CM such that lyl 2 0 and y0 0 (resp.lyl 2 0 y0 < 0).
A restriction of the Penrose correspondence to these spaces gives: So, in the real case we have the following duality: points of M correspond to proSective lines of PN; points of PN correspond to lisht rays M. Note that light rays which can intersect each other in M split into separate points of PN This fact is of the great importance in the twistor theory.
The transformation group SU(2,2) (the group of unitary transformations of T with determinant preserving the form (Z)) preserves the quadric N hence it in- duces transformations of the Minkowski space M which carry light cones again into light cones.In other words, the group SU(2,2) induces conformal transformations of M Moreover, this group is the four-fold covering of the identity component of the conformal group of M (note that elements +/-A, +/-iA of SU(2,2) induce the same transformation of M).Hence, we can define the twistor space (by analogy with the spinors) as a space of the 4-dimensional (fundamental) representation of the sroup SU (2,2) the four-fold covering of the conformal sroup.
It is also interesting to consider the Penrose correspondence for the compactifled Euclidean space E (we recall the Euclidean space E is a subspace of CM where the complex Lorentz metric Izl 2 coincides with Euclidean metric).We have projective line in PT invariant {point of E under mapping j: (Z 0, Z I, Z 2, Z 3) It appears also that a restriction of the Penrose correspondence to E coincides with the natural bundle: CP 3 HP S 4 E i.e.
fibers of this bundle are exactly the j-invariant projective lines in PT or images of points of E under the Penrose correspondence.
It is useful to introduce the Klein model of CM along with the twistor model.
The Klein model can be constructed as follows.We have defined the twistor model of CM as the Grassmann manifold G2(T of two-dimensional subspaces in T. Every such subspace can be defined (up to a non-zero complex multiplier) by a byvector p in A2T given by byvectors e.A e. i i,j 1,2,3,4, where {e i} is a basis of T. Assign to a byvector p six complex numbers {P12' P13' P14' P23' P24' P34 (defined up to proportionality) which are called the PiHcker coordinates of the sub- space.The evident condition p A p 0 is rewritten in the PiHcker coordinates as P12P34 P13P24 + P14P23 0 (2.1) Nence, we have assigned to every plane of G2(T) a point of the projective quadric Q given by (2.1) in CP   5  This is a i-I correspondence and we call the quadric Q the Klein model of Minkowski spac CM.In suitable coordinates it can be written in the form: p + p + p q + q + q The basic objects of the twistor geometry have the following interpretation in terms of the Klein model: x++x+x+x-x--0 lying in Q In 2 we have given a twistor interpretation of geometric objects of CM.How do relativistic fields or solutions of conformally-invariant equations on CM transform under the Penrose correspondence?According to the "twistor programme" of Penrose   ([4]) relativistic fields are to be interpreted in terms of complex geometry of PT, i.e. in terms of holomorphic bundles, cohomologies with coefficients in such bundles and so on.In other words, relativistic equations are "coded" in the complex struc- ture of the twistor space and in this sense equations "disappear" when we pass to twistors.We explain these heuristic considerations first in the case of Maxwell Fabdxa A dx is a 2-form defined by the tensor Fab (a,b =0,1,2,3) of the electromagnetic field, and * is the Hodge operator defined by the metric of M If F has a potential, i.e.F dA for some l-form A, the the equation dF 0 is automatically satisfied, soMaxwell's equation reduce to d(*F) O.We can split the form F into a sum of its self-dual and anti-self-dual components: F F+ + F_ where F+ (iF +/-*F) *F+ +/-iF+ The Maxwell equations in terms of these com- ponents can be written in the form: dF+ O. (If F has a potential and is (anti)- self-dual, i.e.F F+ (or F then Maxwell's equations are automatically satisfied).
In order to obtain a twistor interpretation of the Maxwell equations we must apply the mappings (1.1) and (1.3) to F The tensor Fab transforms under (1.2) a a b where o are Pauli matrices Splitting this into the spinor FAA, BB FaboAA,OBB, spinor into a sum of its self-dual and anti-self-dual components we obtain the spinor version of Maxwell's equations: XAA AB 0 XAA A'B' O, where AB' A'B' are symmetric spinor functions on M. The first equation corresponds to the anti- self-dual Maxwell equation dF_ 0, the second-to the self-dual equation dF+ 0.
More generally, we call the following system LM-equations of spin s: 0 s 0 --i A 0 s 0 XAA AB...L XAA 'B'...L' where AB...L A'B'...L' are symmetric spinor functions on the spinor model of M with 21s spinor indices.Again the first equation is called anti-self-dual, the second-self-dual.The case s +/-I corresponds to Maxwell equations, s +/-2 -linearized EH-equations, s 0 -wave equation.
It is more convenient to begin with a twistor interpretation of holom0r_p_hoi.
solutions of the above equations.Since every distribution solution of LM-equations can be represented as a jump of boundary values of holomorphic solutions in the future and past tube (proved for the wave equation in []), it is sufficient to con- sider the following system of LM-equations in the future tube: where vAA' /ZAA' AB...L' A'B'...L' are symmetric spinor functions (with 21sl indices), holomorphic in the future tube.Note that the spinor model of CM + is a space of 22 -matrices with positive imaginary part.
A twistor interpretation of the anti-self-dual Maxwell equations (equations (3.1)   for s i) is given by the theorem of Penrose ([6]) which asserts that there is a equations in CM + In Dolbeault's representation the cohomology group on the rimht side coincides with a space of smooth -closed (0,1) -forms on PT + modulo -exact forms.The self- dual Maxwell equations have the following interpretation: self-dual holomorphic 1 solutions of Maxwell H (PT+, 0(-4)) equations in CM + where 0(-4) is a sheaf of holomorphic functions of PT whose local sections are given by homogeneous functions of degree -4 in homogeneous coordinates of PT The apparent asymmetry between self-dual and anti-self-dual cases can be removed if we re- place T ir self-dual case by the dual space (PT+) *.Then both formulations will become analogous.
We now explain the idea of the proof of above results in the self-dual case.Let AA' be an element of HI(pT+, 0(-4)).Since a point z e CM + corresponds by (1.3)to a whole projective line in PT+, the value A,B,(Z AA') of pre-images of f under A (1.3) is calculated by averaging f along the projective line: m -izAA A' This averaging is given by a fiber-integral of f along the indicated line with a standard kernel of the form A''B' in homogeneous coordinates [0''I'] on the line.
For the same construction in the anti-self-dual case we need to consider the "normal derivatives" of an element g HI(pT+,0) In other words, we have to go out into an infinitesimal neighbourhood of the line.The value AB(Z AA') of pre-images of g under (1.3)isgivenby a fiber-integral of the "normal derivative" /mA./B (g) The results formulated above for Maxwell equations can be generalized naturally to IM-equations.Namely, the following correspondence solutions of LM-equations with spin s {HI(pT+, 0(-2s-2)) (3.4) in CM + is I-I The mapping (3.4) is constructed in the same way as in the Maxwell case.
There is also a direct twistor interpretation of solutions of LM-equations on M not using their representation as a jump of holomorphic solutions in CM/ To obtain this interpretation one has to replace the cohomologies of PT + in (3.4) by tangent cohomologies of the quadric PN (cf.[7]).A characterization of solutions of LM- equations in terms of (0,2) -flags in PT or incident pairs of points of PT PT* can be given without splitting these solutions into their anti-and self-dual components.
In the case s=0 of a complex wave equation (d'Alembertian) we have a representa- tion of solutions as fiber-integrals of elements of HI(pT+, 0(-2)) along projective lines.
It is interesting to note that an analogue of this representation for the ultra- hyperbolic equation was proved in 1938 by F. John.(Recall that the ultrahyperbolic operator coincides with the restriction of the d'Alembertian to the real subspace of CM where the complex Lorentz metric has signature (2,2).)At the same time this repre- sentation for solutions of the real wave equations and Laplacian, obtained by restric- tion of D'Alembertian to M and E respectively, was apparently unknown.
The YM-equations are matrix analogues of Maxwell's equations.More precisely, consider a principal G -bundle P M with connection over Minkovski space.If a Lie group G is a matrix group, then the connection is given by a matrix valued l-form A A dx a on M with values in the Lie algebra g of G We shall consider a further the case G SU(N) related to the most important physical applications.In this case the coefficients A of the connection are anti-Hermitian N N -matrices a with zero trace.The connection A is called otherwise a matrix potential.The curva- ture of A is given by the covariant derivative F =VAA dA + 1/2[A,A] and is a matrix- (One can check that in the scalar case this theorem coincides with (3.3)).The right side of (4.1) can be redefined as a space of -closed matrix-valued (0,I) -forms modulo exact forms.
(A matrix-valued (0,I) -form u is -closed iff u + [u,u] 0 and is -exact iff u v-lv for some non-degenerate matrix-valued function v).
The idea of the proof is the following.We have a geometric criterion of anti- self-duality: a connection A is anti-self-dual <=> its curvature F vanishes on a -planes.Let A be an anti-self-dual connection in a principal SU(N) bundle P on CM + and E -associated vector bundle.It is convenient to use the following dia- gram: where F + is a space of (0,I) -flags in PT+, i.e. pairs consisting of a point of PT + and a projective line in PT + including this point, and , are natural projec- tions.Denote by E' and A' the trivial liftings of the bundle E and the connec- tion A to F+, and define the bundle E'' on PT + as follows.The fiber of E'' -I over a point Z -PT + is given by A' -horizontal sections of E' over (Z), i.e.
-i by sections s' of E' over (Z) such that VA,S 0 where VA' is the fiber component of VA, along the fibers of v. (This definition is correct due to anti- self-duality of A).The bundle E'' is an image of the field A under (4.1).To prove that this bundle is holomorphic, we define an almost complex structure (Cauchy-VA( ?,l)s,,where s'' is the section of E'' Riemann operator) on E'' by Sdef given by a section s' of the bundle E', and VA(,0'I) is the (0, l)-component of VA,.
To show this definition to be correct it is sufficicent to check that s'' is again V (0'I) V(0,1) a section of E'' i e. that V,, A' s 0 But VA'" A' s VA(?'l)(V,s') + [VA,, VA(,0'I)] s' 0 because V, s' =Oby the given condition, and V V (0'I)] 0 due to anti-self-duality of A. One can show also that 2 A' A' 0 (almost complex analogue of the Frobenius condition) hence by the Newlander-Nirenberg theorem ([9]) this almost complex structure defines, in fact, a complex structure on E''.By construction, the bundle E'' is holomorphically trivial on projective lines in PT + which are images of points of CM+.
Conversely, let E'' be a holomorphic bundle on PT + trivial on projective lines and E' be its trivial lifting to F + Since the fiber of E over a point z CM + is given by holomorplic sections of E' over -1(z).This defines the bundle E on CM + (This definition is correct due to compactness of -l(z) , CpI.)We shall con- struct a connection in E by means of parallel transport.It is sufficient to define it along complex light rays in CM+.The parallel transport in E along a complex light ray is given by identifying fibers of E' over projective lines (corresponding to the different points of the ray with their common point (corresponding to the ray).
The infinitesmial version of this definition defines a connection in E which is anti- self-dual by construction (its curvature is zero on -planes corresponding to the points of PT+).
The self-dual YM-fields have an analogous interpretation in terms of the dual space PT*.An arbitrary YM-field cannot be decomposed into the sum of anti-and self- dual fields owing to nonlinearity of the YM-equations.However, general YM-fields also have a natural twistor interpretation in terms of (0,2)-flags in PT or incident pairs in PT PT* (cf. [10,11]).
The most interesting physical applications of the above results are related to the Euclidean case.The instantons are by definition (anti)-self-dual YM-fields on E which realize minima of the action functional: S(F) By the Ward's theorem there is a I-I correspondence between instantons and holomorphic vector bundles on PT which are holomorphically trivial on j-invariant projective lines (cf. 2) and have an additional quaternionic structure generated by With the help of this re- suit there was given in [12] a description of the space of instantons in terms of quaternion matrices, satisfying some quadratic constraints.Considering physical appli- cations of this result we note that it would be desirable to have (if possible) a more explicit description of the space of instantons because one has to integrate over this space for quantisation.
There is also one more important class of (anti)-self-dual solutions of YMequations called monopoles.They are by definition (anti)-self-dual YM-fields A (A) on E not depending on "time" (i.e.coordinate x) and satisfying the a boundary condition: ]I A II o This condition realizes minima of the energy functional: E(F) f(F,*F)dxldx2dx3.
It is natural to consider the monopoles as fields, so called Yang-Mills-Higgs fields, on the 3-dimensional Euclidean space E The space T(CP I) (the tangent bundle space of the Riemann sphere) appears to be an analogue of the twistor space PT in this case.We have again an analogue of Ward's theorem ( [13]) which states that there is a I-1 correspondence between Yang-Mills-Higgs fields on E and holomorphic vector vector bundles on T(CP I) which is holomorphically trivial on projective lines in T(CP I) and has an addio-al quaternionic structure.Hitchin [13] used this theorem to construct for every monopole some algebraic curve (called a spectral curve) thus reducing the problem of description of monopoles to the description of such curves.It is interesting to note that a similar construction appeared in the paper by K.
Weirstrass in 1866 dedicated to the minimal surfaces in E The twistor approach surfaces it is ani-self-dual by the criterion.Conversely, an anti-self-dual manifold M has a 3-complex-parameter collection of a-surfaces and the manifold of these sur- faces is identified with T To define a metric on M i.e. to obtain a complex solution of EH-equations, we have to introduce according to the "twistor programme" of Penrose some additional structure on T not belonging to the "complex geometry" of For instance let be a sufficiently small neighbourhood of a rational curve L in PT Then we can identify T with the normal bundle of L and other rational curves in with holomorphic sections of this bundle.Denote also by K the canonical line bundle of 3-forms on T (written in local coordinates in the form f dz Adz 2 Adz ).Then a restriction of KI to L coincides with the standard bundle 0(-4) on CP The theorem of Penrose ([14]) asserts that these data are sufficient for the construction of anti-self-dual solutions of EH-equations.More precisely, there is (locally) a i-I correspondence holomorphic bundles : CP with 4-parameter collection of {anti-self-dual manifolds} sections and isomorphism *0 KT (-4) Unfortunately, this result cannot be applied to the construction of real Lorentz solu- tions of EH-equations because real anti-self-dual manifolds always have even signature.
Nevertheless, this theorem can be successfully applied to the construction of Euclidean anti-self-dual manifolds which are considered in quantum gravity.To this class belong ALE (asymptotically locally Euclidean)-manifolds introduced in [16] which have topology of S3/F R in a neighbourhood of infinity where F is a finite group of isometrics on S The twistor interpretation of these spaces in kind of (I0) was given in [17].
We want to underline first a connection between Penrose transformation and another interesting mathematical result theorem of Sato-Kawai-Kashiwara ([18]).By this theorem an arbitrary overdetermined system of pseudodifferential equations in general position can be transformed microlocally (i.e.locally in the cotangent bundle) by means of a canonical transformation of infinite order into a system of tangential Cauchy-Riemann equations.This transformation (which does not reduce in general to a coordinate transformation in the base space) preserves only analytic singularities of solutions.The Penrose transform also carries the field theory equations into systems of tangential Cauchy-Riemann equations on twistor manifolds and, moreover, it allows one to obtain explicit formulas for solutions.
There is a close connection between the twistor approach and the method of Riemann- Hilbert boundary, problem (cf.[19]) which is also applied to solution of the self-dual YM-equations.The solution of these equations by the indicated method reduces to solving the factorisation problem for a rational matrix-valued function on the Riemann sphere CP (which is equivalent to a trivialization of a ho!omorphic bundle on CP defined by this matrix function) depending on three complex parameters.This is equiva- lent to a construction of a holomorphic vector bundle on PT holomorphically trivial on projective lines which are images of points on CM The Penrose transform is attached to 4-dimensional manifolds.Its existence on the group-theoretical language is due to the following local group isomorphlsms: S0(4) S0(3) S0(3) (note that the group S0(n) is simple for n 3, n # 4), SL(4,C) 0(6,C) SU(2,2) S0(4,2) All these isomorphisms are "fasten" to the dimension four.

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12 tity
2x2 -matrix, and consider the two-dimensional plane in C 4 which is generated by the basis consisting of the two four-dimensional columns of this matrix.The space two-dimensional subspaces in C 4 i.e. the Grassmann manifold G2(C 4), is of all called the twistor model of the complex Minkowski spa.ce and is denoted by CM The mapping constructed above assigns to every matrix Z CM a point of the space CM We shall write this mapping in coordinates.Denote coordinates in C 4 by Z (Z) (ZO,Z I,Z2,Z3) and consider a vector Z C 4 as a pair of spinors Z (A,A,), i.e.

valued 2 -
form: F Fabdxa A dx b Fab a bAa + [Aa'Ab] where /x a a [A ,] is the commutator of matrices A A connection A is called a YM(Yang X a a Mills) -field if its curvature satisfies the YM -equation: VA(*F) 0 <=> d(*F) + [A,*FI 0 A field A is (anti)-self-dual if *F (-)iF (Anti)-self-dual fields automatically satisfy the YM-equations (by the Bianchi identity).I,et us describe first a twistor interpretation of holomorphic anti-self-dual YM -fields on CM + By the theorem of Ward ([8]) there is a I-I correspondence: Figol

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